Plethystic Murnaghan-Nakayama rule via vertex operators

Based on the vertex operator realization of the Schur functions, a determinant-type plethystic Murnaghan--Nakayama rule is obtained and utilized to derive a general formula of the expansion coefficients of $s_{\nu}$ in the plethysm product $(p_{n}\circ h_{k})s_{\mu}$. Meanwhile, the equivalence between our algebraic rule and the combinatorial one is also established. As an application, we provide a simple way to compute the generalized Waring formula.